The conformal geometry of spacetime in four and higher dimensions is controlled by
the Weyl tensor. The necessary and sufficient condition for spacetime to be conformally flat is that the Weyl tensor is equal to zero, see e.g. [1]}.
If conformal gravity is realised as the gauge theory of the conformal group [2]},
then the algebra of conformally covariant derivatives is determined by the Weyl tensor and
its covariant derivatives [3]}, [4]}, [5]}. The conformal connection becomes flat if the Weyl tensor vanishes.
